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Operator Splitting and the Parareal Algorithm ∗ Jürgen Geiser A, , Stefan Güttel B

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Operator Splitting and the Parareal Algorithm ∗ Jürgen Geiser A, , Stefan Güttel B View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector J. Math. Anal. Appl. 388 (2012) 873–887 Contents lists available at SciVerse ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Coupling methods for heat transfer and heat flow: Operator splitting and the parareal algorithm ∗ Jürgen Geiser a, , Stefan Güttel b a Humboldt University of Berlin, Unter den Linden 6, D-10099 Berlin, Germany b University of Oxford, England, United Kingdom article info abstract Article history: We propose an operator splitting method for coupling heat transfer and heat flow Received 25 May 2011 equations. This work is motivated by the need to couple independent industrial heat Available online 3 November 2011 transfer solvers (e.g., the Aura-Fluid software package) and heat flow solvers (e.g., Submitted by W.L. Wendland Openfoam). Such packages are often used to simulate the influence of solar heat in car bodies and are coupled by A-B splitting techniques. The main goal of this work is the Keywords: Heat transfer acceleration of the coupled software system by iterative operator splitting methods and Heat flux additional time-parallelism using the parareal algorithm. We present these new splitting Operator splitting techniques along with some novel convergence results and test the splitting-parareal Iterative operator splitting combination on various numerical problems. Parareal © 2011 Elsevier Inc. All rights reserved. 1. Introduction We are interested in the fast solution of coupled heat transfer and heat flow equations with splitting and time paral- lelization techniques. We concentrate on iterative splitting methods, which are related to iterative solver methods and have two historical origins: • Picard–Lindelöf iterations [13,14], where waveform relaxation methods have their origins [20]. • Iterative split-operator approaches, see e.g. [10], wherein a two step method is derived to solve nonlinear reactive transport equations. A detailed description along with the theoretical background of iterative splitting schemes is given in the monograph [8]. The main advantage of iterative splitting schemes is the possibility to obtain higher accuracy of order O(τ i ) with addi- tional iterative steps (where i denotes the number of iterations and τ is the length of the time-step). This is different to standard splitting schemes with fixed order, e.g. A-B splitting of order O(τ ) or Strang splitting of order O(τ 2) (see [18]), where an increased accuracy can only be achieved through smaller time-steps τ , with the cost of an increased number of function evaluations, see [17]. Another advantage of (iterative) splitting schemes compared to the simultaneous solution of the full equation system is the possible simplification of the splitted equations representing decomposed operators, for which specialized solvers can be employed. For some practical problems, splitting techniques are unavoidable due to the fact that decoupled software packages specialized on different physical problems need to be combined. In the industrial application we have in mind (heat in car bodies), we have two different equations being solved with separate codes: while the heat transfer and radiation * Corresponding author. E-mail addresses: [email protected] (J. Geiser), [email protected] (S. Güttel). 0022-247X/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2011.10.030 874 J. Geiser, S. Güttel / J. Math. Anal. Appl. 388 (2012) 873–887 is solved with the Aura software package, the flow field of the temperature is computed with a flow-field solver, e.g. Openfoam or Vectis, see [19]. These software packages also implement space parallelization by domain decomposition. However, in order to employ even more processors (with a number above the critical number of processors for which space parallelization is saturated) we make use of the so-called parareal iteration, see [15]. This paper is organized as follows. The general model is described in Section 2. The splitting methods related to the model equations are given in Section 3. In Section 4 we briefly review the parareal time-parallelization iteration used in Section 5, wherein we report on various numerical experiments with test problems. 2. Mathematical model We consider a model of coupled heat transfer and heat flow. The heat transfer problem arises, for example, when the temperature distribution in a car body needs to be modeled over time, see [21], and therefore we have to apply large time-steps to simulate such problems. The heat flow is given as a simplification of the inviscid compressible Navier–Stokes equation, see [16]. We deal with a inviscid flow based on the assumption that viscous forces are small in comparison to inertial forces. Such situations can be identified as flows with a Reynolds number much greater than one. The assumption that viscous forces are negligible can be used to simplify the Navier–Stokes equations to the Euler equations. 1. Heat transfer equation (Heat equation): ∂t T =∇·(K ∇T ) −∇·(vT ), T (x,t0) = T0(x), (2.1) where the unknown temperature is denoted by T . This equation contains a diffusion part and a convection part, both of which are typically treated as independent operators in splitting methods. 2. Heat flow equation (Euler equation): ∂t v =−(v ·∇)v −∇p(T ), v(x,t0) = v0(x), (2.2) where the unknown flux is v and we assume ∇·v = 0. This equation contains a nonlinear flow part and a pressure part, both of which are typically treated as different, independent operators in splitting methods. However, in this paper we are not mainly concerned with decomposing the equations themselves, but rather in the coupling of both equations to each other by an iterative splitting scheme, see [12]. Remark 2.1. For simplicity, we sometimes (e.g., in the first of our numerical experiments) assume that p is independent of T , in which case we obtain a uni-directional coupling between (2.2) and (2.1). This means that one can solve (2.2) and then use the result directly for solving (2.1). Remark 2.2. The coupling of both equations becomes more involved when the dependence between them is strong or highly nonlinear. In particular, even the evaluation of ∇ p(T ) can become difficult if the pressure is only given implicitly. Therefore the idea is to apply an iterative splitting scheme, which couples the two delicate equations via a relaxation scheme, see [7]. With this as an advantage, we can do better than a non-iterative scheme that has stability problems, while we deal with stiff problems, see [17]. Here we use the smoothing property of the iterative schemes and overcome the stiffness problem, see [8]. 3. Splitting methods In the following we briefly discuss the coupling methods used for the heat transfer equation. 3.1. Lie–Trotter or A-B splitting method The simplest scheme (and probably the one implemented most often) is the so-called A-B splitting method: n n+1 n n ∂t v =−(v ·∇)v, with t t t , v x,t = v (x), (3.1) ∂ T + + =∇·(K ∇T ) −∇·v˜ T , with tn t tn 1, v˜ x,tn = vn 1(x), T x,tn = T x,tn (3.2) ∂t n n+1 n+1 where T is the known initial value of the previous solution and T (t ) = T2(x,t ) is the approximate solution of the full equation. This method results in a global splitting error O (t), where t denotes the length of the time-step. J. Geiser, S. Güttel / J. Math. Anal. Appl. 388 (2012) 873–887 875 3.2. Strang splitting The Strang splitting method is given by the following equations: ∂ T1(x,t) n n+1/2 n n n n =∇·(K ∇T1) −∇·v1 T1, with t t t , v1 x,t = v (x), T1 x,t = T x,t , (3.3) ∂t n n+1 n n ∂t v2 =−(v2 ·∇)v2, with t t t , v2 x,t = v (x), (3.4) ∂ T3(x,t) =∇·(K ∇T3) −∇·v3 T3, ∂t n+1/2 n+1 n+1/2 = n+1 = n+1/2 with t t t , v3 x,t v2 (x), T3(x,tn+1/2) T1 x,t , (3.5) n n+1 n+1 where T is the known initial value of the previous solution and T (t ) = T3(x,t ) is the approximate solution of the full equation. This method has a global splitting error of order O (t2). 3.3. Iterative splitting method 3.3.1. Linear case The following algorithm is based on an iteration with fixed splitting discretization step-size τ [1]. On the time interval + [tn,tn 1] we solve the following subproblems consecutively for i = 0, 2,...,2m: n n+1 n n ∂t vi =−(vi ·∇)vi, with t t t , vi x,t = v (x), (3.6) ∂ Ti n n+1 n n n n =∇·(K ∇Ti) −∇·vi−1 Ti, with t t t , vi−1 x,t = v (x), T x,t = T x,t (3.7) ∂t where T n, vn is the split approximation at time t = tn [1]. Here we solve the time-discretization with a backward differential formula of fourth order (BDF4 method), see [9]. The higher order is obtained by applying recursively a fixed-point iteration to reconstruct the analytical solution of the coupled operators, see [5]. 3.3.2. Generalization to systems of ODEs We deal with a vectorial iterative scheme, given as an inner and outer iterative scheme. While the outer iterative scheme is a Waveform relaxation scheme and could be seen as a coarse iterative scheme, since we iterate over the full system in one step.
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